Please Help: A Survey on the Mathematical Experience

Luke Wolcott is a recent PhD graduate and previous member of the editorial board of this blog. He is currently working on a postdoc project and could use our help. Please see Luke’s note below.

As a side project during my research postdoc, I’ve been collaborating with a philosopher of mathematics, Alexandra Van Quynh, on mathematical phenomenology. We’re looking for help from math PhD students (or postdocs, or professors). We’ve decided to look at the mathematician’s lived experience of groups (the abstract algebra kind). We’re mainly interested in the mental imagery, metaphors, and narratives that mathematicians use to understand and work with groups. Our working assumption is that this internal experience differs significantly from the formal manifestations of groups that appear in math literature, that it is subjective but that there are commonalities among mathematicians. The motivation came in part from this MathOverflow post (started by the late Bill Thurston), in part from Lakoff and Nunez’s Where Mathematics Comes From, in part from a survey of working mathematicians that (mathematician!) Jacques Hadamard used in the 1950s as data to conjecture patterns of mathematical intuition. I think groups are a great test object, because they’re basic enough that all mathematicians are familiar with them, they show up in so many different places in mathematics, and they’re perhaps the first step into abstract algebra. Groups are one of the first, most basic math objects that are (relatively) severed from the real world, and in particular visual or spacial intuition. Well, I don’t want to give away our hypotheses too much. We’ve made a survey, asking questions about the lived experience of groups, and we’re distributing it to mathematicians. Would you consider completing the survey? The survey is below. Spend as much or little time on it as you feel like. It’s not necessary to overthink your answers; sometimes the first, spontaneous response is the more revealing. We may get back to you and ask some follow-up questions about your responses. We’ll keep your responses confidential, or we’ll ask your permission if we decide it would be nice to quote you. Enjoy!

Name

Email

Please choose a group — any group that comes to your mind naturally — and answer the following nine questions about working with that group. You may refer to one specific instance, or speak in general terms.

1) What is the group you are experiencing now?(required)

2) In what mathematical context do you use or work with this group?

3) Does the group look like anything? If yes, please describe any images that come to your mind. Any colors, shapes, or textures? In what dimensions?

4) More generally, does this group evoke any sounds or other sensations?

5) Can you describe the process of working with this group? For instance, how do you mentally manipulate it?

6) Does the group have a personality? If so, please give details.

7) Is this group part of a story? If so, please give details.

8) With regards to the aspects discussed above, how has your internal experience of this group changed over time?

9) Are you aware of which aspects of this internal experience were taught to you and which you developed yourself?

The next two questions are about the symmetric group.

10) How do you think about the symmetric group?

11) What is the essence of the symmetric group?

Lastly, a few bonus questions:

12) What does it mean to understand a group? What does it mean to not understand a group?

13) What is your favorite group axiom? Why?

About Tyler Clark

I am a second year doctoral student in mathematics at the University of Central Florida in Orlando, FL. I obtained my MS from WKU in Bowling Green, KY where I researched continued radicals and Cantor sets.